Frequency interval balanced truncation of discretetime bilinear systems

Transcription

1 Jazlan et al., Cogent Engineering 206, 3: SYSTEMS & CONTROL RESEARCH ARTICLE Frequency interval balanced truncation of discretetime bilinear systems Ahmad Jazlan,2 *, Victor Sreeram, Hamid Reza Shaker 3 and Roberto Togneri Received: 29 March 206 Accepted: 0 June 206 First Published: 20 June 206 *Corresponding author: Ahmad Jazlan, School of Electrical, Electronics and Computer Engineering, University of Western Australia, 35 Stirling Highway, Crawley, Perth, Western Australia 6009, Australia; Faculty of Engineering, Department of Mechatronics Engineering, International Islamic University Malaysia, Jalan Gombak, 5300 Kuala Lumpur, Malaysia ahmadjazlan@iium.edu.my Reviewing editor: James Lam, University of Hong Kong, Hong Kong Additional information is available at the end of the article Abstract: This paper presents the development of a new model reduction method for discrete-time bilinear systems based on the balanced truncation framework. In many model reduction applications, it is advantageous to analyze the characteristics of the system with emphasis on particular frequency intervals of interest. In order to analyze the degree of controllability and observability of discrete-time bilinear systems with emphasis on particular frequency intervals of interest, new generalized frequency interval controllability and observability gramians are introduced in this paper. These gramians are the solution to a pair of new generalized Lyapunov equations. The conditions for solvability of these new generalized Lyapunov equations are derived and a numerical solution method for solving these generalized Lyapunov equations is presented. Numerical examples which illustrate the usage of the new generalized frequency interval controllability and observability gramians as part of the balanced truncation framework are provided to demonstrate the performance of the proposed method. Subjects: Dynamical Control Systems; Non-Linear Systems; Systems & Control Engineering Keywords: model reduction; bilinear systems; balanced truncation; frequency interval gramians; finite frequency interval Ahmad Jazlan ABOUT THE AUTHORS Ahmad Jazlan is currently a PhD student at the School of Electrical, Electronics and Computer Engineering, University of Western Australia. Victor Sreeram is currenty a professor at the School of Electrical, Electronic, and Computer Engineering, University of Western Australia. He is on the editorial board of many journals including IET Control, Theory and Applications, Asian Journal of Control, and Smart Grid and Renewable Energy. Hamid Reza Shaker is currently an associate professor at the Center for Energy Informatics, University of Southern Denmark. Roberto Togneri is currently a professor at the School of Electrical, Electronic and Computer Engineering. He is currently an associate fditor for IEEE Signal Processing Magazine Lecture Notes and IEEE Transactions on Speech, Audio and Language Processing. This research work is applicable to both engineering and mathematical problems which can be formulated as bilinear systems. PUBLIC INTEREST STATEMENT Nonlinear mathematical models are commonly used to describe the processes in many branches of engineering. Bilinear systems are an important class of nonlinear systems which have wellestablished theories and are applicable to many practical applications. Mathematical models in the form of bilinear systems can be found in a variety of fields such as the mathematical models which describe the processes of electrical networks, hydraulic systems, heat transfer, and chemical processes. Many nonlinear systems can be modeled as bilinear systems with appropriate state feedback or can be approximated as bilinear systems by using the bilinearization process. The mathematical modeling of a large-scale bilinear system may result in a high-order bilinear model. To address the complexity associated with highorder models, we present a new model reduction technique for discrete-time bilinear systems. 206 The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY 4.0 license. Page of 5

2 Jazlan et al., Cogent Engineering 206, 3: Introduction Model reduction which is of fundamental importance in many modeling and control applications deals with the approximation of a higher order model by a lower order model such that the input output behavior of the original system is preserved to a required accuracy. The balanced truncation model reduction technique originally developed by Moore for continuous-time linear systems is one of the most widely applied model reduction techniques Moore, 98. In recent years, many variations to this original balanced truncation technique have been developed Li, Yu, Gao, & Zhang, 204; Minh, Battle, & Fossas, 204; Opmeer & Reis, 205; Zhang, Wu, Shi, & Zhao, 205. One of the further developments to the original balanced truncation technique was the work by Gawronski and Juang which involved the development of frequency interval controllability and observability gramians Gawronski & Juang, 990. The significance of emphasizing particular frequency intervals of interest in a variety of control engineering problems has led to extensive theoretical developments in robust control techniques which emphasize particular frequency intervals of interest which have been presented in Ding, Du, & Li, 205; Ding, Li, Du, & Xie, 206; Du, Fan, & Ding, 206; Imran & Ghafoor, 205; Li & Yang, 205; Li, Yin, & Gao, 204; Li, Yu, & Gao, 205. In the context of discrete-time systems, digital systems are designed to work with signals with known frequency characteristics, therefore it is essential to have model reduction techniques which generate reduced-order models which function well with signals which have specified frequency characteristics. The works by Horta, Juang, and Longman 993, Wang and Zilouchian 2000 and more recently by Imran and Ghafoor 204 described the formulation of frequency interval gramians for discrete-time systems. Bilinear systems are an important category of non-linear systems which have well-established theories Al-Baiyat, Bettayeb, & Al-Saggaf, 994; Dorissen, 989; D Alessandro, Isidori, & Ruberti, 974; Shaker & Tahavori, 204b, 205. Many non-linear systems in various branches of engineering can be well represented by bilinear systems. Similar to the case of linear systems, the mathematical modeling process to obtain bilinear system models may result in obtaining high-order models. Fortunately, by formulating a state space model for these bilinear system models, the application of model reduction techniques becomes possible to reduce the order of these bilinear system models. The balanced truncation technique for continuoustime bilinear systems has been presented in Zhang & Lam 2002 as the balanced truncation technique for discrete-time bilinear systems has been presented in Zhang, Lam, Huang, and Yang More recently further developments have been carried out to the original balanced truncation technique for continuous-time bilinear systems in order to reduce the approximation error between the outputs of the original bilinear model and reduced-order bilinear model by incorporating time and frequency interval techniques Shaker & Tahavori, 204a, 204c. The contributions of this paper are as follows. Firstly, new generalized frequency interval controllability and observability gramians are defined for discrete-time bilinear systems. Secondly, it is shown that these frequency interval controllability and observability gramians are solutions to a pair of new generalized Lyapunov equations. Thirdly, conditions for solvability of these new generalized Lyapunov equation are proposed together with a numerical solution method for solving these new Lyapunov equations. Finally, numerical examples are provided to demonstrate the performance of the proposed method relative to existing techniques. Page 2 of 5

3 Jazlan et al., Cogent Engineering 206, 3: The notation used in this paper is as follows. M refers to the transpose of the matrix M if M R n m and complex conjugate transpose if M C n m. The symbol denotes a Kronecker product. 2. Preliminaries 2.. Controllability and observability gramians of discrete-time linear systems Considering the following time-invariant and asymptotically stable discrete-time linear system A, B, C: xk + Axk+ Buk yk Cxk u R p, y R q, C R n are the input, output and states respectively. A R n n, B R n p, C R q n are matrices with appropriate dimensions. Definition The discrete-time domain controllability and observability gramian definitions are given by: P A k BB A k 2 k0 Q A k C CA k 3 k0 Remark It is established that 2 and 3 satisfy the following Lyapunov equations: APA P + BB 0 A QA Q + C C Remark 2 By applying a direct application of Parseval s theorem to 2 and 3, the controllability and observability gramians in the frequency domain are given by: P 2π Q 2π π π π π e jθ I A BB e jθ I A dθ e jθ I A C Ce jθ I A dθ I is an identity matrix Frequency interval controllability and observability gramians of discrete-time linear systems Definition 2 The frequency interval controllability and observability gramians for discrete-time systems are defined as Horta et al., 993: P cf e jθ I A BB e jθ I A dθ Q cf e jθ I A C Ce jθ I A dθ [θ, θ 2 is the frequency range of operation and 0 θ <θ 2 π. Due to the symmetry of the discrete Fourier transform, the integration is carried out throughout the frequency intervals [θ, θ 2 and [ θ 2, θ Horta et al., 993. Therefore the gramians P cf and Q cf in 8 and 9 will always be real. 8 9 Page 3 of 5

4 Jazlan et al., Cogent Engineering 206, 3: Remark 3 It has been shown that the frequency interval controllability and observability gramians defined in 8 and 9 are the solutions to the following Lyapunov equations Wang et al., 2000: AP cf A P cf + X cf 0 A Q cf A Q cf + Y cf 0 0 X cf F cf BB + BB F cf Y cf F cf C C + C CF cf 2 3 and F cf θ θ 2 I + I Ae jθ dθ 2π Controllability and observability gramians of discrete-time bilinear systems Considering the following discrete-time bilinear system represented by: m xk + Axk+ xku j k+buk yk Cxk 5 6 xk R n n is the state vector, uk R m m is the input vector and u j k is the corresponding jth element of uk, yk R q q is the output vector and A, B, C and are matrices with suitable dimensions. This bilinear system is denoted as A,, B, C. The controllability gramian for this system is defined as Zhang et al., 2003: P P i P i i k i 0 k 0 7 P k A k B P i k,...k i A i[ k N P i N 2 P i N m P i, i 2 as the observability gramian is defined as Zhang et al., 2003: Q Q Q i i i k i 0 k 0 8 Q k CA k Q i k,...k i Q i N Q i N 2 Q i N m The controllability and observability gramians defined in 7 and 8 are the solution to the following generalized Lyapunov equations Zhang et al., 2003: Page 4 of 5

5 Jazlan et al., Cogent Engineering 206, 3: APA P + PN j + BB 0 A QA Q + N QN + j C C The generalized Lyapunov equations corresponding to the controllability and observability gramians in 9 and 20 can be solved iteratively. The controllability gramian can be obtained by Zhang et al., 2003: P lim i Pi 2 A P A P i + BB 0, A P i A P + Pi N j + BB 0, i 2 as the observability gramian can be obtained by Zhang et al., 2003 Q lim i Qi A Q A Q i + C C 0, A Qi A Q + Q i + C C 0, i 2 3. Main work N j Frequency interval controllability and observability gramians of discrete-time bilinear systems For a particular discrete-time frequency interval Ω [γ, γ 2, we define the frequency interval controllability and observability gramians as follows: Definition 3 The generalized frequency interval controllability gramian for discrete-time bilinear systems is defined as: Pθ: Pi θ 2π i,..., θ i P θ,..., θ dθ dθ i i i i 25 [γ, γ 2 and P θ e jθ I A B [ P i θ,..., θ i e jθ i I A N Pi N 2 Pi N m Pi Similarly, the generalized frequency interval observability gramian is defined as: Qθ: i 2π i Q i θ,..., θ i Q i θ,..., θ i dθ dθ i 26 Page 5 of 5

6 Jazlan et al., Cogent Engineering 206, 3: [γ, γ 2 and Q θ Ce jθ I A Q i θ,..., θ i N Qi N 2 Qi N m Qi e jθ i I A These gramians defined in 25 and 26 are the solution to a pair of new generalized Lyapunov equations which is presented in Theorem. Lemmas, 2 and 3 together with Theorem presented in the following sections are interrelated such that Lemma and Lemma 2 are required as part of proving Lemma 3, as Lemma 3 is required for proving Theorem. Lemma Let A be a square matrix which is also stable and let M be a matrix with the appropriate dimension. If X satisfies the following equation: X + io A i MA i 27 It follows that X is the solution to: AXA X + M 0 Proof Since X + i0 Ai MA i and A is stable, it follows that: 28 + AXA X A A i MA i A i i0 + Since + i0 Ai+ MA i+ + i Ai MA i AXA A i0 + i A i+ MA i+ A i MA i + i0 + i0 + i0 A i MA i A i MA i A i MA i M. Lemma 2 Let A be a square matrix which is also stable and let R be a matrix with the appropriate dimension. If Y satisfies the following Y + io A i RA i It follows that Y is the solution to A YA Y + R Proof Similar to the proof of Lemma and is therefore omitted for brevity. Page 6 of 5

7 Jazlan et al., Cogent Engineering 206, 3: Lemma 3 Let M and R be matrices with the appropriate dimensions and let A be stable, if P cf and Q cf satisfy: P cf e jθ I A Me jθ I A dθ Q cf e jθ I A Re jθ I A dθ 3 32 then P cf and Q cf are the solution to the following generalized Lyapunov equations: A P cf A P cf X cf A Qcf A Q cf Y cf X cf FM MF Y cf F R RF and F θ θ 2 I + θ 2 I Ae jθ dθ... 2π 2π θ 2 θ I Ae jθ dθ 2π θ 37 Proof In this part we will prove that 3 is the solution to 33. This proof is a further development of the proof of equation 4.a in Wang and Zilouchian The proof that 32 is the solution to 34 can then be obtained similarly by using lemma 2 and therefore is omitted for brevity. Firstly 28 can be re-written as follows: e jθ I AXe jθ I A +Xe jθ I A e jθ I... +e jθ I AXe jθ I M 38 Multiplying 38 from the left by e jθ I A and from the right by e jθ I A followed by integrating both sides by dθ yields: 2π e jθ I A Me jθ I A dθ Xdθ + e jθ I A Xe jθ dθ +... Xe jθ e jθ I A θ Xdθ + I e jθ IA dθ X +... X I e jθ IA dθ 39 Page 7 of 5

8 Jazlan et al., Cogent Engineering 206, 3: Denoting K I e jθ IA dθ, 39 can be re-written as: P cf Xdθ + K X + XK 40 Substituting 40 into the left-hand side of 33 yields: [ A Xdθ + K 2π X + XK A... [ Xdθ + K 2π X + XK X cf 4 It has been shown in Wang and Zilouchian 2000 that the property AK K A and AK K A holds true. As a result 4 can be re-written as X cf dθi[axa X+K 2π [AXA X+[AXA XK dθi M K 2π M MK dθi + K 4π M M dθi + K 4π is equivalent to the right-hand side of 35. By comparing both expressions we have F 4π dθi + K 43 Due to the symmetry of the discrete Fourier transform, the integrations are carried out throughout the frequency intervals [θ, θ 2 and [ θ 2, θ Horta et al., 993. Therefore we have F θ θ 2 I + θ 2 I Ae jθ dθ... 2π 2π θ 2 θ I Ae jθ dθ 2π θ Lemma 3 derived in the previous section is now applied as part of the proof of Theorem as follows. Theorem The frequency interval controllablity and observability gramians Pθ and Qθ defined in 25 and 26 are the solutions to the following generalized Lyapunov equations: m A PθA Pθ+F PθN j +... m PθN j F + FBB + BB F 0 m A QθA Qθ+F N j Qθ +... m Qθ F + F C C + C CF 0 N j Page 8 of 5

9 Jazlan et al., Cogent Engineering 206, 3: Proof The proof that the frequency interval controllability gramian Pθ defined in 25 is the solution to the generalized Lyapunov equation in 44 is presented in this section. The proof that the frequency interval observability gramian Qθ defined in 26 is the solution to the generalized Lyapunov equation in 45 can be obtained in a similar manner and therefore is omitted for brevity. Firstly let: P θ P θ P θdθ P i θ Pi θ, θ 2,..., θ i P i θ, θ 2,..., θ i dθ...dθ i we have Pθ P i θ i Using Lemma 3 with M BB, it is observed that P θ is the solution to 48 A P θa P θ+fbb + BB F 0 49 For P 2 θ we have P 2 θ P2 θ 2π i, θ 2 P θ, θ dθ dθ e jθ I A [N 2π i P N m P... P N P N m m e jθ I A dθ dθ 2 e jθ I A... P θ 2π P θdθ N j e jθ I A dθ 2 m e jθ I A P θn j e jθ I A dθ 2 Denoting M m P θn j, Lemma 3 applies and as a result P 2 θ will be the solution to: m A P 2 θa P m 2 θ+f P θn j + P θn j F 0 50 Similarly, according to Lemma 3, P i θ will be the solution to m A P i θa P m i θ+f Pi θn j + Pi θn j F 0 5 Page 9 of 5

10 Jazlan et al., Cogent Engineering 206, 3: Adding 5 to 49 and applying a summation to infinity as in the right-hand side of 48 yields m A P i θa P i θ+f P i θn j +... i i i2 m 52 P i θn j F + FBB + BB F 0 i2 Equivalently, we have m A P i θa P i θ+f P i θn j +... i i m P i θn j F + FBB + BB F 0 i i Finally applying the property in 48 to 53 m A PθA Pθ+F m PθN j + PθN j F + FBB + BB F Conditions for solvability of the Lyapunov equations corresponding to frequency interval controllability and observability gramians In this section, the condition for solvability of the generalized Lyapunov equation in 44 which corresponds to the frequency interval controllability gramian defined in 25 is presented herewith in Theorem 2. The condition for solvability of the generalized Lyapunov equation in 45 which corresponds to the frequency interval observability gramian defined in 26 can be derived in a similar manner and is therefore omitted for brevity. Theorem 2 The generalized Lyapunov equation in 44 is solvable and has a unique solution if and only if 53 W A A I I+ F +F 54 is non-singular. Proof Let vec. be an operator which converts a matrix into a vector by stacking the columns of the matrix on top of each other. This operator has the following useful property [20 vecm, M 2, M 3 M 3 M vecm 2 55 Applying vec. on both sides of 44 together with the property in 54 yields { m m A A I I+ F + F }vec Pθ vecfbb + BB F 56 The generalized Lyapunov equation in 44 is solvable provided that this equation presented in 56 is solvable and has a unique solution. It follows that 56 is solvable and has a unique solution if and only if Page 0 of 5

11 Jazlan et al., Cogent Engineering 206, 3: W A A I I+ F +F is non-singular Numerical solution method for the Lyapunov equations corresponding to the frequency interval controllability and observability gramians The iterative procedure for solving bilinear Lyapunov equations in previous studies can also be applied to obtain the solution to the generalized Lyapunov equation in 44 - Pθ as follows Shaker et al., 204a, 204c; Zhang et al., 2003; Zhang & Lam, 2002: Pθ lim i + Pi θ 57 A P θa P θ+fbb + BB F 0 m A P i θa P i θ+f Pi θn j +... m Pi θn j F + FBB + BB F 0, i This iterative procedure can also be applied to solve the generalized Lyapunov equation corresponding to the frequency interval observability gramian in Model reduction algorithm The procedure for obtaining the reduced-order model is described as follows Step : The frequency interval controllability and observability gramians are calculated by solving 44 and 45, respectively. Step 2: Both of these frequency interval controllability and observability gramians obtained by solving 44 and 45 are simultaneously diagonalized by using a suitable transformation matrix denoted by T such that T PθT T T T QθT Step 3: Transform and partition to get the realization [ [ Ā T A A AT 2, N T N N NT 2 A 2 A 22 N 2 N 22 [ B T B [ B, C CT C C 2 B 2 Step 4: The reduced order model is given by A r A, N r N, B r B, C r C 4. Results and discussion 4.. Numerical example and results Considering the following fifth-order discrete-time bilinear system originally presented by Hinamoto and Maekawa 984 which has also been used by Zhang et al Page of 5

12 Jazlan et al., Cogent Engineering 206, 3: Table. Exact values of yk from Figure k yk Original Proposed Zhang et al E E m xk + Axk+ xku j k+buk yk Cxk A C [ , B The proposed technique involves firstly obtaining the frequency interval controllability and observability gramians defined in Theorem and subsequently using these gramians as part of the balanced truncation-based technique described in Section 3.4 Moore, 98; Zhang & Lam, 2002; Zhang et al., This fifth-order model {A, N, B, C} is reduced to the following second-order model {A r, N r, B r, C r } in the form of 60 by using the proposed technique for the frequency interval Ω[0.04π, 0.3π, N Similarly, by applying the proposed technique for the frequency interval Ω [0, 0.π to this fifthorder model {A, N, B, C}, a third-order discrete-time bilinear system with the following system matrices {A r2, N r2, B r2, C r2 } in the form of 60 is obtained [ A r C r [ [.2986, B r [ , N r Table 2. Exact values of yk from Figure 2 k yk Original Proposed Zhang et al E E Page 2 of 5

13 Jazlan et al., Cogent Engineering 206, 3: A r C r2 [ B r , N r For comparison, we apply the method by Zhang et al which yields the following second- and third-order discrete-time bilinear systems with the following system matrices {A r3, N r3, B r3, C r3 } and {A r4, N r4, B r4, C r4 } in the form of 60 [ A r [ N r and A r4 N r [.3335, B r Discussion of results Figure shows the step responses of the original fifth-order model, second-order model obtained using the proposed method for a frequency interval Ω [0.04π, 0.3π and a second-order model obtained using the method by Zhang et al Table shows the exact values for yk for the discrete times k 20, 2, 22, 23, 24, 25 from Figure. E and E2 denote the absolute error between the value of yk of the original model and the value of yk obtained using the proposed method and the method by Zhang et al. 2003, respectively. On the other hand Figure 2 shows the step responses of the original fifth-order model, thirdorder model obtained using the proposed method for a frequency interval Ω [0, 0.π and a third-order model obtained using the method by Zhang et al Table 2 shows the exact values for yk for the discrete times k 20, 2, 22, 23, 24, 25 from Figure 2. E and E2 denote the absolute error between the value of yk of the original model and the value of yk obtained using the proposed method and the method by Zhang et al. 2003, respectively. From both,, C r3 [ , B r , C r4 [ Figure. Step responses of the original model, second-order reduced model obtained using the method by Zhang et al and second-order reduced model obtained using the proposed technique for a frequency Ω[0.04π, 0.3π. yk Original Model L. Zhang et al 2003 Proposed Method k Page 3 of 5

14 Jazlan et al., Cogent Engineering 206, 3: Figure 2. Step responses of the original model, thirdorder reduced model obtained using the method by Zhang et al and third-order reduced model obtained using the proposed technique for a frequency Ω [0, 0.π. yk Original Model L. Zhang et al 2003 Proposed Method k Figure, Figure 2, Table, and Table 2 it is shown that applying the proposed technique yields a reduced-order model which is a closer approximation to the original model compared to the method by Zhang et al Conclusion In conclusion, a new model reduction method for discrete time bilinear systems based on balanced truncation has been developed. The frequency interval controllability and observability gramians for discrete time bilinear systems are introduced and are shown to be solutions to a pair of new generalized Lyapunov equations. The conditions for solvability of these new Lyapunov equations are provided and the numerical solution method used to solve these equations is explained. Numerical results show that the proposed method yields reduced-order models which is a closer approximation to the original model as compared to existing techniques. The technique proposed in this paper is applicable to a variety of non-linear systems which can be formulated as bilinear systems. Funding The authors received no direct funding for this research. Author details Ahmad Jazlan,2 ahmadjazlan@iium.edu.my Victor Sreeram victor.sreeram@uwa.edu.au Hamid Reza Shaker 3 hrsh@sdu.dk Roberto Togneri roberto.togneri@uwa.edu.au School of Electrical, Electronics and Computer Engineering, University of Western Australia, 35 Stirling Highway, Crawley, Perth, Western Australia 6009, Australia. 2 Faculty of Engineering, Department of Mechatronics Engineering, International Islamic University Malaysia, Jalan Gombak, 5300 Kuala Lumpur, Malaysia. 3 Center for Energy Informatics, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark. Citation information Cite this article as: Frequency interval balanced truncation of discrete-time bilinear systems, Ahmad Jazlan, Victor Sreeram, Hamid Reza Shaker & Roberto Togneri, Cogent Engineering 206, 3: References Al-Baiyat, S. A., Bettayeb, M., & Al-Saggaf, U. M New model reduction scheme for bilinear systems. International Journal of Systems Science, 25, D Alessandro, P., Isidori, A., & Ruberti, A Realization and structure theory of bilinear dynamical systems. SIAM Journal on Control, 2, Ding, D., Du, X., & Li, X Finite-frequency model reduction of two-dimensional digital filters. IEEE Transactions on Automatic Control, 60, Ding, D. W., Li, X. J., Du, X., & Xie, X Finite-frequency model reduction of takagi-sugeno fuzzy systems. IEEE Transactions on Fuzzy System, PPPP, 0. Dorissen, H Canonical forms for bilinear systems. Systems and Control Letters, 3, Du, X., Fan, F., & Ding, D Finite-frequency model order reduction of discrete-time linear time-delayed systems. Nonlinear Dynamics, XX, 2. Gawronski, W., & Juang, J Model reduction in limited time and frequency intervals. International Journal of System Science, 2, Hinamoto, T., & Maekawa, S Approximation of polynomial state-affine discrete time systems. IEEE Transactions on Circuits and Systems, 3, Horta, L., Juang, J., & Longman, R Discrete-time model reduction in limited frequency ranges. Journal of Guidance, Control, and Dynamics, 6, Page 4 of 5

15 Jazlan et al., Cogent Engineering 206, 3: Imran, M., & Ghafoor, A Stability preserving model reduction technique and error bounds using frequency limited gramians for discrete-time systems. IEEE Transactions on Circuits and Systems - II: Express Briefs, 6, Imran, M., & Ghafoor, A Model reduction of descriptor systems using frequency limited gramians. Journal of the Franklin Institute, 352, Li, X., & Yang, G Adaptive control in finite frequency domain for uncertain linear systems. Information Sciences, 34, Li, X., Yin, S., & Gao, H Passivity-preserving model reduction with finite frequency approximation performance. Automatica, 50, Li, X., Yu, C., & Gao, H Frequency Limited H Model Reduction for Positive Systems. IEEE Transactions on Automatic Control, 60, Li, X., Yu, C., Gao, H., & Zhang, L A New Approach to h model reduction for positive systems. Proceedings of the 9th IFAC World Congress. Cape Town. Minh, H. B., Battle, C., & Fossas, E A new estimation of the lower error bound in balanced truncation method. Automatica, 50, Moore, B. 98. Principal component analysis in linear system: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control, AC-26, Opmeer, M., & Reis, T A lower bound for the balanced truncation error for mimo systems. IEEE Transactions on Automatic Control, 60, Shaker, H., & Tahavori, M. 204a. Frequency interval model reduction of bilinear systems. IEEE Transactions on Automatic Control, 59, Shaker, H. & Tahavori, M. 204b. Generalized hankel interaction index array for control structure selection for discrete-time mimo bilinear processes and plants pp Proceedings of the 53rd IEEE Conference on Decision and Control CDC. Los Angeles, CA. Shaker, H., & Tahavori, M. 204c. Time-interval model reduction of bilinear systems. International Journal of Control, 87, Shaker, H., & Tahavori, M Control configuration selection for bilinear systems via generalised hankel interaction index array. International Journal of Control, 88, Wang, D., & Zilouchian, A Model reduction of discrete linear systems via frequency-domain balanced structure. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47, Zhang, H., Wu, L., Shi, P., & Zhao, Y Balanced truncation approach to model reduction of markovian jump timevarying delay systems. Journal of the Franklin Institute, 352, Zhang, L., & Lam, J On H 2 model reduction of bilinear systems. Automatica, 38, Zhang, L., Lam, J., Huang, B., & Yang, G. H On gramians and balanced truncation of discrete-time bilinear systems. International Journal of Control, 76, The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY 4.0 license. You are free to: Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Cogent Engineering ISSN: is published by Cogent OA, part of Taylor & Francis Group. Publishing with Cogent OA ensures: Immediate, universal access to your article on publication High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online Download and citation statistics for your article Rapid online publication Input from, and dialog with, expert editors and editorial boards Retention of full copyright of your article Guaranteed legacy preservation of your article Discounts and waivers for authors in developing regions Submit your manuscript to a Cogent OA journal at Page 5 of 5